Ed Hellen's Web Page
Interesting Dynamics (aka Nonlinear Dynamics)
Chaotic
systems: Real-time
bifurcation diagrams produced from analog electronic circuits. Simple
quadratic equations used in finite-difference recursion relations display
a wide variety of behaviors including chaos. Click
for article (pdf) or Edward
H. Hellen, Am. J. Phys. 72, 499(2004).
Control of Chaos: Deterministic chaos (as produced by the above examples) can be controlled by small perturbations of a system parameter (R or a). We derive the approach to stability and verify it experimentally for a delayed feedback control method. Graph shows measurements of system values and control parameter taken from controlled Henon electronic circuit.
Here is an arXiv Eprint: J.K. Thomas and E.H. Hellen
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In addition to stabilizing a single fixed point, it is also possible to stabilize unstable periodic orbits. Here we show numerical simulation of control of Period-2 oscillation. System values are red. Perturbations to control parameter are green. Control was turned on at 50, off at 200, back on at 250.
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Excitable Systems: Fitzhugh-Nagumo.
Here are a couple of VPython avi movies. Fitzhugh-Nagumo1
(shows phase-space), Fitzhugh-Nagumo2
(shows phase-space and time evolution). The system includes both a steady
leakage current and a pulsed stimulation. Here is a spatiotemporal version showing
propagation of excitations too close together: Fitz-Nag_Diffusion
Modeling Excitable Systems, arXiv Eprint: J.L. Lancaster and E.H. Hellen Describes electronic and mathematical models.
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Nonlinear
Damping of the LC Circuit using Anti-parallel Diodes. Interesting amplitude
dependent behavior occurs when anti-parallel diodes replace the resistor in
a RLC circuit. The circuit becomes a nonlinearly damped harmonic oscillator
obeying the equation:
where 
More info.
or get article Hellen and Lanctot,
Am. J. Phys. 75, 326 (2007) Or click for pdf
Logarithmic
decay:
Diode-capacitor circuit. What happens when the resistor
in an RC decay circuit is replaced by a diode? The decay changes from
exponential to logarithmic. (At least to a good approximation for 5
decades of time.) Click for more on Diode-Capacitor
Decay. For reprint of article, Click here
(pdf) or, this article may also be found at Edward
H. Hellen, Am. J. Phys. 71, 797(2003).
Padé-Laplace and Signal Averaging
Pade-Laplace is an intriguing method for finding the decay
constants in a multi-exponential decay. Unlike standard methods of curve-fitting,
it does not use the sum of the square of the errors and does not assume the
number of decay constants. Instead it uses Laplace transforms, Pade approximants,
Taylor series expansion, matrix inversion, and finding the zeroes of polynomials
in order to determine the number of decay constants, their values, and their
amplitudes.
Here Padé-Laplace analysis is used on signal averaged data
obtained from a simple circuit that produces multi-exponential voltage decays
in the presence of noise. Click for article
(pdf) or Edward H. Hellen,
Am. J. Phys. 73, 871 (2005).
Other Stuff
Here is Classic Beauty, perfection that probably will never be repeated.
| Fun Electronics (No semiconductors here)
6L6, Thordarson,
Eico
De La Rive Electric Discharge Tube (fig at right) in action qv_x_B.
To save the qv x B movie, right-click here and "Save Target as" using extension .wmv instead of .html. |
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Total Internal Reflection Fluorescence Microscopy
TIRFM (avi of VPython animation). This one shows
the evanescent illumination: TIRFM2. These movies demonstrate how TIR is achieved through the microscope
objective. See publication of early use
of this method to measure hormone (epidermal growth factor) binding
kinetics at cell surface. (Part of my Ph.D. thesis)
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Nearby hiking Pic1
Pic2 Pic3 Mov1
AAPT Summer 05 Salt Lake City Pic 1 Pic
2 Pic 3 |
Python notes here
Intro Physics python demos here |
| Publications |
UNCG Physics |
